Entropy in Markov analysis of late paleozoic cyclical coal measures of East Bokaro basin, Bihar, India

Randomness in the occurrence of lithologies in a cyclical succession is evaluated in terms of entropies which can be calculated from a Markov chain matrix. Two types of entropies are linked with every lithologic state; one is the entropy before deposition E ^(pre) and the other is that after deposition E ^(post),which together form an entropy set. The entropy sets for pebbly sandstone, sandstone, shale, and coal for the Karharbari coal measures, and sandstone, shale, carbonaceous shale, and coal for the Barakar coal measures were plotted separately and compared with Hattori's idealized plots. These coal measures probably were essentially of symmetrical cyclical pattern (Type-B)of Hattori. The entropy of the whole sedimentation unit readily fits under the broad framework of fluvial cycles.     

Markov chain analysis applied to an ancient alluvial plain succession

Markov chain analysis is a comparatively simple statistical technique for the detection of repetitive processes in space or time. Coal measure cyclothems or fluvial fining-upward cycles are good examples of sedimentary successions laid down under the control of Markovian processes. Analyses of stratigraphic sections commence with a transition count matrix, a two-dimensional array in which all possible vertical lithologic transitions are tabulated. Various probability matrices may be derived from this raw data, and these are then subjected to chi-square tests to determine the presence or absence of the Markov property. This technique is applied to four types of stratigraphic succession which occur in the Devonian rocks of Prince of Wales Island, Arctic Canada. (1) A conglomerate succession of alluvial fan origin. Markov analysis is of little or no assistance in the interpretation of these rocks, in which only two principal lithologies are present. (2) A conglomerate-sandstone succession. Fluvial fining-upward cycles are detectable by visual examination of the sections and are strongly indicated by Markov analysis. (3) A sandstone-carbonate succession, of marginal marine origin, and including both marine and non-marine strata. Cyclicity is weak in these rocks, but analysis suggests that regressions took place much more rapidly than transgressions during their period of deposition. (4) A succession in which the relative proportions of the various lithologies vary markedly with age. The varying nature of the cyclic tendencies is emphasized in this case by dividing the succession into two subintervals, for the purpose of analysis.     

Cyclic characteristics of coal-bearing sediments in the Bochumer Formation (Westphal A 2) Ruhrgebiet, Germany

Cyclic characteristics of the Bochumer coal measures have been studied statistically using first-order embedded Markov chain analysis on data from thirty-two boreholes. Separate tally matrices which were made for the Lower, Middle and Upper units by summing the tally matrices for all the boreholes, were used to produce various transition probability matrices, which were then subjected to the chi-square test. Results strongly suggest that the Bochumer succession was deposited by a Markovian mechanism, and that the sequence represents cyclic sedimentation. The preferential upward transition path for lithologic changes that can be derived in the total area and in individual logs is: coal\rightarrow shale\rightarrowsiltstone\rightarrowsandstone\rightarrowsiltstone\rightarrowrooty bed\rightarrowcoal. Among the noteworthy features are a two-way transition between siltstone and sandstone, which implies frequent interbedding of the two lithologies, and the transition, up out of siltstone to rooty beds. There are, however, minor variations, locally, either by way of subsidiary transitions or in the extent to which the above transitions can be followed. When tested for stationarity through time and space, the Markov chain analysis yields results which show the upward sequence of lithologic states (facies) to be stationary at individual localities, but non-stationary when the entire data are tested for the whole area. The relationship of the number of coal-bearing cycles to total thickness of strata and average thickness of coal-cycles is examined using linear regression. Significant positive relationships are present between the number of cycles and total thickness, and significant negative correlations occur between the number of cycles and their average thickness. The results of the study have been used to infer sedimentological evolution of the succession and sediment cyclicity. Environmentally, the Bochumer cycles fit suitably into the concept of deltaic cycles.     

Markovian models in stratigraphic analysis

A stratigraphic section may be divided into lithologic units which in turn may be divided into beds. This paper gives a mathematical formulation of stratigraphic sections that takes these two levels into account and uses bed properties to yield the thickness and number of beds in lithologic units. The model is a semiMarkov chain in which the succession of lithologic bed types forms a Markov chain and is an independent random variable. The model is tested against stratigraphic data obtained from micrologs. There is close agreement between the observed and calculated thicknesses of lithologic units. Tests for the degree of agreement between observed and calculated numbers of beds in lithologic units are hampered by inability to observe thin beds on micrologs. Some implications of this limitation to stratigraphic analysis are noted.     

岩相剖面的定量历史沉积学分析

岩相剖面的定量历史沉积学分析是以动力沉积学分析为基础、以数学地质为方法、以计算机为工具的定性-定量分析法.它主要包括马尔科夫链模型中的转移概率矩阵分析,置换分析,极限概率和马尔科夫链的首达时间分析,熵分析和时间序列分析,本文侧重从地质学的角度阐述了它们的原理和方法。     

Application of Markov chain and entropy analysis to lithologic succession — an example from the early Permian Barakar Formation,Bellampalli coalfield,Andhra Pradesh,India

A statistical approach by a modified Markov process model and entropy function is used to prove that the early Permian Barakar Formation of the Bellampalli coalfield developed distinct cyclicities during deposition. From results, the transition path of lithological states typical for the Bellampalli basin is as: coarse to medium-grained sandstone → interbedded fine-grained sandstone/shale → shale → coal and again shale. The majority of cycles are symmetrical but asymmetrical cycles are present as well. The chi-square stationarity test implies that these cycles are stationary in space and time. The cycles are interpreted in terms of in-channel, point bar and overbank facies association in a fluvial system. The randomness in the occurrence of facies within a cycle is evaluated in terms of entropy, which can be calculated from the Markov matrices. Two types of entropies are calculated for every facies state; entropy after deposition E(post) and entropy before deposition E(pre), which together form entropy set; the entropy for the whole system is also calculated. These values are plotted and compared with Hattori’s idealized plots, which indicate that the sequence is essentially a symmetrical cycle (type-B of Hattroi). The symmetrical cyclical deposition of early Permian Barakar Formation is explained by the lateral migration of stream channels in response to varying discharge and rate of deposition across the alluvial plain. In addition, the fining upward cycles in the upper part enclosing thick beds of fine clastics, as well as coal may represent differential subsidence of depositional basin.     

The accuracy of standard enthalpies and entropies for phases of petrological interest derived from density-functional calculations

The internal energies and entropies of 21 well-known minerals were calculated using the density functional theory (DFT), viz. kyanite, sillimanite, andalusite, albite, microcline, forsterite, fayalite, diopside, jadeite, hedenbergite, pyrope, grossular, talc, pyrophyllite, phlogopite, annite, muscovite, brucite, portlandite, tremolite, and CaTiO₃–perovskite. These thermodynamic quantities were then transformed into standard enthalpies of formation from the elements and standard entropies enabling a direct comparison with tabulated values. The deviations from reference enthalpy and entropy values are in the order of several kJ/mol and several J/mol/K, respectively, from which the former is more relevant. In the case of phase transitions, the DFT-computed thermodynamic data of involved phases turned out to be accurate and using them in phase diagram calculations yields reasonable results. This is shown for the Al₂SiO₅ polymorphs. The DFT-based phase boundaries are comparable to those derived from internally consistent thermodynamic data sets. They even suggest an improvement, because they agree with petrological observations concerning the coexistence of kyanite?+?quartz?+?corundum in high-grade metamorphic rocks, which are not reproduced correctly using internally consistent data sets. The DFT-derived thermodynamic data are also accurate enough for computing the P–T positions of reactions that are characterized by relatively large reaction enthalpies (>?100 kJ/mol), i.e., dehydration reactions. For reactions with small reaction enthalpies (a few kJ/mol), the DFT errors are too large. They, however, are still far better than enthalpy and entropy values obtained from estimation methods.     

Cyclic sedimentation of the Barren Measures Formation (Damuda Group), Talchir Gondwana basin: Statistical appraisal from borehole logs

The succession of lithofacies of a part of the Barren Measures Formation of the Talchir Gondwana basin has been studied by statistical techniques. The lithologies have been grouped under five facies states viz coarse-, medium-, and fine-grained sandstones, shale and coal for statistical analyses. Markov chain analysis indicates the arrangement of Barren Measures lithofacies in the form of fining upward cycles. A complete cycle consists of conglomerate or coarse-grained sandstone at the base sequentially succeeded by medium-and fine-grained sandstones, shales and coal at the top. The entropy analysis puts the Barren Measures cycles into A-4 type cyclicity, which consists of different proportions of lower, upper and side truncated cycles of lithologic states. Regression analysis indicates a sympathetic relationship between total thickness of strata (net subsidence) and number of cycles and an antipathic relationship between average thickness and number of sedimentary cycles. The cyclic sedimentation of the Barren Measures Formation was controlled by autocyclic process which occurred due to the lateral migration of streams triggered by intrabasinal differential subsidence. In many instances, the clastic sediments from the laterally migrating rivers interrupted the cyclic sedimentation resulting in thinner cycles in areas where the number of cycles are more. Principal component and multivariate regression analyses suggest that the net subsidence of the basin is mostly controlled by the thickness of sandstones, shale beds and coal stringers.     

Microbialite morphostratigraphy as a tool for correlating Late Cambrian-Early Ordovician sequences

Microbialite morphostratigraphy is a new tool for intrabasinal correlation using diverse microbialite structures (morphotypes). The recognition of the succession of morphotypes over constrained temporal intervals and broad areas is a function of the complex interactions that operate to create the structure. Because so many nonlinked variables (e.g., biotic, sedimentological, physicochemical) are involved, similar morphotypes do not reoccur over long temporal intervals. To demonstrate the technique, the upper Cambrian-lowermost Ordovician shelf strata of the Great Basin, United States, were correlated using both morphostratigraphy and standard lithostratigraphy. Six morphozones and one morphosubzone were recognized, as were four main lithologic successions. Because the boundaries between the morphozones and lithologic successions did not coincide, it is inferred that the characteristics of the various microbialite structures are not solely controlled by physical factors. The principles for establishing a morphostratigraphy outlined in this article allow for the potential to correlate along other ancient marine margins in both the same Cambrian and Ordovician interval, as well as any interval in the Phanerozoic in which diverse microbialite structures occur.     

Basic Lithofacies-Succession Model for the Wumishan Cyclothems: Their Markov Chain Analysis and Regularly Vertical Stacking Patterns in the Third-Order Sequences

The Mesoproterozoic Wumishan Formation in the Jixian section of Tianjin is a succession of 3300-m-thick carbonate strata formed in a period of about 100 Ma (1310±20 Ma-1207±10 Ma). In this succession of strata, the carbonate metre-scale cyclic sequences belonging to peritidal type with an approximately symmetrical lithofacies-succession are best developed. The wide development of 1:4 stacking patterns shows that these metre-scale cyclic sequences are genetically related to the short-eccentricity cycles, which are called the Wumishan cyclothems that could truly represent sedimentary cycles. Generally, massive and thick-bedded calcareous dolomites and dolomitic limestones of stromatolite biostromes and thrombolite bioherms constitute the central part of the Wumishan cyclothems. The lower and upper parts consist of tidal flat dolostones, sandy-muddy dolostone and the top part is composed of lagoonal facies dolomitic shales with a paleosol cap. Therefore, an approximately symmetrical lithofacies-successio     

Patterns of spatial distribution of ground successions in the territory of Russia

A scheme for the spatial distribution of ground successions in the territory of Russia is constructed. Peculiarities of the ground composition and structure in the section of the succession, the character of the state of ground successions caused by the phase state of water in the successions, annual mean temperatures and degree of humidity for thawed and non-frozen rocks, as well as the degree of ice content for permafrost rocks, were used as qualification indicators. The variety of recognized ground successions, which differ in their genesis and specific features of structure, composition, and state, are demonstrated. Patterns of their distribution are discussed.     

潜江盐湖盆地生储盖组合特征

潜江盐湖盆地是我国内陆独一无二的高盐度盐湖沉积盆地。本文在分析其岩性韵律特征的基础上,探讨了潜江盐湖生储盖组合特征。研究表明,潜江凹陷潜江组是在干湿频繁交替的古气候条件下,在高盐度、强蒸发、还原—强还原水体中,由北部单向碎屑物源及凹陷周缘卤水与盐源补给形成的盐系地层,岩性组合的有序变化形成了多套生储盖组合系统,不但反映出潜江盐湖沉积的特殊性和复杂性,而且具有十分优越的成油气地质条件。     

Genetic Types of Meter-Scale Cyclic Sequences and Fabric Natures of Facies Succession

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High-Frequency Sequence Stratigraphy Using Syntactic Methods and Clustering Applied to the Upper Limestone Coal Group (Pendleian, E1) of the Kincardine Basin, United Kingdom

We define a distance between sedimentary successions to compare their dissimilarity formally. Distance definition is based on attributed syntactic representation. One-dimensional successions can be represented by a string of lithofacies symbols sequentially or vertically. Each symbol can also have a vector of attributes that can provide other information on lithofacies such as thickness. The distance of any two successions is then defined consisting of its syntactic and attribute subdistances. Syntactic distance measures difference of vertical lithofacies change between two successions and attribute distance measures difference of thickness of corresponding lithofacies. Clustering is used to test validity of distance definition and its potential application to analysis of cycle-dominated sedimentary successions. Example is from the Namurian-A succession in Kincardine basin, central Scotland. There are 56 cycles in intervals of about 300 m each in two boreholes. Recognition of intermediate cycles depends on correctly determining of types of these short cycles and their vertical stacking pattern. Intermediate cycles have better potential in high-resolution stratal correlation regionally. Syntactic clustering results show that 56 short cycles can be classified into four groups with distinctive geological interpretation, which further helps reveal hierarchical cyclic architecture of the whole succession.     

环流分解及其在沉积地层旋回分析中的应用

本文应用马尔柯夫链进行地层序列分析。当一个地层序列形成一个具有马尔柯夫性质和极限概率分布的马尔柯夫链时,它的转移概率矩阵可分解为细致平衡部分和环流部分。根据环流矩阵可确定该序列中的主要沉积旋回趋势,为地层序列分析提供重要依据。本文给出了两个研究实例。     

Markov chains and embedded Markov chains in geology

Geological data are structured as first-order, discrete-state discrete-time Markov chains in two main ways. In one, observations are spaced equally in time or space to yield transition probability matrices with nonzero elements in the main diagonal; in the other, only state transitions are recorded, to yield matrices with diagonal elements exactly equal to zero. The mathematical differences in these two approaches are reviewed here, using stratigraphic data as an example. Simulations from chains with diagonal elements greater than zero always yield geometric distributions of lithologic unit thickness, and their use is recommended only if the input data have the same distribution. For thickness distributions lognormally or otherwise distributed, the embedded chain is preferable. The mathematical portions of this paper are well known, but are not readily available in publications normally used by geologists. One purpose of this paper is to provide an explicit treatment of the mathematical foundations on which applications of Markov processes in geology depend.     

沉积旋回叠置形式的波形分析及旋回层序划分方法

在陆源碎屑岩地层的沉积记录中,测井曲线上一个砂泥岩旋回可以采用正弦波进行描述,测井曲线幅度的高低反映旋回粒度大小的变化,旋回波长则是度量旋回厚度的标尺。本研究表明,测井曲线变化幅度与其平均值之差计算的累积残差曲线,是识别沉积体系、判别旋回层序界面的一个有效图解方法。根据数值模拟实验,证实对于任意一个级次的旋回复合波系,累积残差曲线的正半波和负半波曲线分别对应低频旋回的向上变细和向上变粗的沉积序列,正半波和负半波曲线的转换点位置对应旋回层序界面的深度。同时,引进测井曲线的频谱分析和滤波方法,可以划分沉积旋回的级别,进而研究沉积旋回变化形式的驱动机制。文中以柴达木盆地东部地区仙3井为例,根据测井数据计算的累积残差曲线,说明了划分低频和高频旋回层序的方法和流程,讨论了旋回幅度、波长变化与沉积环境和沉积速率的关系。     

上扬子区寒武系的层序地层格架——寒武纪生物多样性事件形成背景的思考

上扬子区,特别是贵州及其邻区的寒武系发育完整,是一套从黑色页岩系到碳酸盐岩的地层序列,该序列构成一个复杂而有序的层序地层格架。在时间变化方面,寒武系由两个二级层序组成,滇东统和黔东统所组成的二级层序可以进一步划分为5个三级层序,武陵统和芙蓉统所组成的二级层序可以进一步划分为7个三级层序,最终形成一个有序的海侵—海退旋回序列;在滇东统和黔东统所构成的二级层序之中,三级层序形成一个有序的垂直叠加形式:滇东统和黔东统自下而上三级层序由“凝缩段(CS)+高水位体系域(HST)”序列演变为“海侵体系域(TST)+凝缩段(CS)+高水位体系域(HST)”,层序界面类型由淹没不整合型层序界面演变为明显的暴露间断面,意味着二级层序的相序组构与三级层序存在相似性;武陵统和芙蓉统所构成的二级层序之中,厚度巨大而且化石贫乏的娄山关群白云岩自北西向南东的进积尖灭,表明该二级层序为一个总体向上变浅序列,而且其中的大多数三级层序不发育凝缩段单元,与滇东统和黔东统形成强烈的反差。在空间变化方面,从北西向南东随着古地理背景由浅变深,滇东统和黔东统厚度变薄,向东南变为一个难以进行三级层序划分的凝缩序列;武陵统和芙蓉统则从浅水台地相白云岩的向南东相变为一套陆棚相泥页岩和泥灰岩为主的地层序列。研究区域寒武系的层序地层格架不但表明了三级层序的两大属性——“空间上相序的有序性和时间上环境变化的同步性”,而且也反映出地层记录中“两种相变面和两种穿时性”。在层序地层框架内可以观察到:在第一个二级海侵作用期间,从小壳化石组合经松林生物群到与澄江动物群同时代的遵义生物群,形成了寒武纪早期最为引人注目的生物多样性事件;在第二个海侵作用期间,从台江生物群到凯里生物群,又形成了另一个显著的生物多样性事件。该事实表明,大规模的海侵作用为生物多样性事件奠定了基本的沉积背景,而海退事件是否对应着生物灭绝事件还有待于进一步研究。因此,海侵—海退事件与生物变化事件之间的复杂关系还需要更进一步的研究才能得到更加合理的阐释。     

Applications of maximum entropy and optimal number of class interval concepts: Two examples

Two data sets, one from surface sediment samples obtained from a subtidal sand wave and another consisting of three wave-height records representing different oceanographic conditions, are employed to test the application of maximum entropy (ME) and optimal number of class interval (K) concepts. Each data set was modified further to obtain a frequency distribution determined by eight unequal-size class intervals. Both original and modified sets were treated by a reliable statistical method. Comparison of relative entropies and results from statistical treatments show the advantage of the transformation of any original data set by means of ME and K methods before analyzing it further.Contribution 127, Instituto Argentino de Oceanografia.     

Analysis of Vertical Sedimentary Successions in the Lower Part of Kelimoli Formation,Middle Ordovician,Zhuozishan Area

There is significance to study high-frequency cyclicity thin-bedded limestone which deposited in carbonate ramp and deep-water basin for its well record of astronomic cycles. And there is an ideal section for these studies in the lower part of Kelimoli Formation, in Darriwilian Stage, Middle Ordovician of western Ordos Basin(Zhuozishan area in Inner Mongolia) for its well rhythmic alternations of thin-bedded limestones and mudstones. This work focused on detailed vertical sedimentary successions analysis of these thin-bedded limestones based on detailed field work, as well as the data of rare earth elements and other related trace elements in limestones. The results show as follows: ① there are four types of sedimentary successions(bundles) in thin-bedded limestones, including thinning-upward succession, thickening-upward succession, bidirectional succession(thickening- and thinning-upward) and waving-upward succession; ② the section consists of 145 limestone-mudstone rhythmics, 33 bundles and 7 superbundles; ③ the variation in(La/Nd)_N and Mn/Fe molar ratios of limestones along with depth have a similar trend which decreases from bottom to top in general, but have some anomalies distribution; ④ the variation in REE concentrations and Eu anomalies along with depth have a perfect enantiomorphous relationship as well as coupling with the molar ratios of(La/Nd)_N; ⑤ the variation of crystal size, V/Cr and V/(V+Ni) molar ratios also have a similar trend which is increasing and then decreasing from bottom to top, and have some local enantiomorphous relationship with the molar ratios of(La/Nd)_N in the overall similar distribution patterns. Some conclusion can be drawn according to the types of sedimentary successions and its distribution in profile as well as the ratios of some geochemical parameter along depth based on previous works. These include: ① thickening-upward and bidirectional successions maybe reflect the intensity of tectonism and contour current which is controlled by Milankovitch cycle respectively, and the rising of sea level may be was responsible both for the thinning-upward and waving-upward successions; ② the intensity of contour current which is controlled by Milankovitch cycle was acted throughout the depositions, and the range of deposition time is not less than 2.9 Ma; ③ the vertical sedimentary evolution is perhaps controlled by the alternating of tectonism and rising sea levels, and the section suggests two tectonic stages.